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arXiv · 2606.25641

Towards Hodge-Riemann relations for non-Archimedean analogs of valuations on convex sets

Semyon Alesker
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Abstract

In [8], a non-Archimedean analogue of the space of translation-invariant even valuations on convex sets was introduced. In [7], motivated by a further analogy with the classical theory, this space was equipped with two multiplicative structures, the product and the convolution. Both structures satisfy Poincare duality and the (non-mixed) hard Lefschetz theorem. In this paper, we formulate a conjecture concerning a more general mixed versions of the hard Lefschetz theorem and the Hodge-Riemann relations. We prove the non-mixed Hodge-Riemann relations in degree 1 for the product and, equivalently, in codegree 1 for the convolution.

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Semyon Alesker. 2026-06-24. Towards Hodge-Riemann relations for non-Archimedean analogs of valuations on convex sets. https://arxiv.org/abs/2606.25641

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